# Characteristic subgroup of finite index

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: characteristic subgroup and subgroup of finite index

View other subgroup property conjunctions | view all subgroup properties

## Definition

A **characteristic subgroup of finite index** is a subgroup of a group that is a characteristic subgroup as well as a subgroup of finite index, i.e., the index of the subgroup is finite, or equivalently, the quotient group is a finite group.

## Relation with other properties

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Normal subgroup of finite index | ||||

Characteristic core of subgroup of fininte index | ||||

Subgroup of finite index | Normal subgroup of finite index|FULL LIST, MORE INFO | |||

Characteristic subgroup | Characteristic core of subgroup of finite index, Powering-invariant characteristic subgroup, Quotient-powering-invariant characteristic subgroup|FULL LIST, MORE INFO |